Statistical mechanics 2f) 



after a passage are transformed into the variables 

 (35) u^, r\, u^, v.,, lo^ 



we have evidently to identify the variables (33) and the variables (34), so that we have 



(in) = AQ M I As/ Ae./ r7'f dh' b' // 



In this formula we substitute the variables (35) for the variables (34). As, 

 according to the invariant theorem of the preceding paragraphs, 



fj, w^; ^^2, v^, 

 we thus get 



(in) = A9.Mf Asj As^ r/'l dh h' c/ // /g' , 



where instead of fc'w' may be written èco. 



In this integral Mj, v^, ii\ are to be kept unaltered, so that As^^ may be placed 

 before the sign of integration, and the integration is to be performed over all values 

 of , «2 , «f 2 between — co and -\- œ . 



According to (8*) we now get 



(in) - (out) = Mi M As, /■ (// 72' - /; f,) A£2 d'^dhho^ ■ 



(36) 



= AQA^As, V(/), 



which formula gives the increase in the number of stars within the dominion As, 

 caused by passages during the time A^. 

 We hence have 



(37) V{f)=j\\jj{J\' A' -fuQ^^^^hd'^<^'^>^ 



where 



Asg = cZmj dv.^ div^ 

 and the integration is to be performed over all values of 



']j between 0 and 2t^, 

 h » d >y \ B, 

 u^, v^, tv^ * — ^ » GO . 

 I remind of the fact that 



f^' = f{x, ?/, z\ t; w/), 



so that we have to introduce in (37) instead of m/, y/, tv^'; u^', v^', h\' their values 

 through i\, w^; «2» ^2' ^■'^2 given in the formulae (18). 



Lnnds Universitets Årsskrift. N. F. Avd. 2. Bd 28. 



4 



